Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries
نویسنده
چکیده
By “elementary” plane geometry I mean the geometry of lines and circles—straightedge and compass constructions—in both Euclidean and non-Euclidean planes. An axiomatic description of it is in Sections 1.1, 1.2, and 1.6. This survey highlights some foundational history and some interesting recent discoveries that deserve to be better known, such as the hierarchies of axiom systems, Aristotle’s axiom as a “missing link,” Bolyai’s discovery—proved and generalized by William Jagy—of the relationship of “circle-squaring” in a hyperbolic plane to Fermat primes, the undecidability, incompleteness, and consistency of elementary Euclidean geometry, and much more. A main theme is what Hilbert called “the purity of methods of proof,” exemplified in his and his early twentieth century successors’ works on foundations of geometry.
منابع مشابه
Spatial Analysis in curved spaces with Non-Euclidean Geometry
The ultimate goal of spatial information, both as part of technology and as science, is to answer questions and issues related to space, place, and location. Therefore, geometry is widely used for description, storage, and analysis. Undoubtedly, one of the most essential features of spatial information is geometric features, and one of the most obvious types of analysis is the geometric type an...
متن کاملThe Evolution of Urban Zoning from Conventional to Form Based Codes; Introducing Non-Euclidean Zoning Techniqueschniques
Zoning has always been one of the basic tools of land use control available. Zoning is the regulation and restriction of land uses according to a predetermined plan. This paper will present a look at conventional zoning, its origins, the evolution of the zoning, and the scope of zoning types. Regardless of the varieties, most zoning codes can be classified into at least one of the following bro...
متن کاملNon-euclidean geometries: the Cayley-Klein approach
A. Cayley and F. Klein discovered in the nineteenth century that euclidean and non-euclidean geometries can be considered as mathematical structures living inside projective-metric spaces. They outlined this idea with respect to the real projective plane and established (“begründeten”) in this way the hyperbolic and elliptic geometry. The generalization of this approach to projective spaces ove...
متن کاملمدلسازی صفحهای محیطهای داخلی با استفاده از تصاویر RGB-D
In robotic applications and especially 3D map generation of indoor environments, analyzing RGB-D images have become a key problem. The mapping problem is one of the most important problems in creating autonomous mobile robots. Autonomous mobile robots are used in mine excavation, rescue missions in collapsed buildings and even planets’ exploration. Furthermore, indoor mapping is beneficial in f...
متن کاملAssessment of the Log-Euclidean Metric Performance in Diffusion Tensor Image Segmentation
Introduction: Appropriate definition of the distance measure between diffusion tensors has a deep impact on Diffusion Tensor Image (DTI) segmentation results. The geodesic metric is the best distance measure since it yields high-quality segmentation results. However, the important problem with the geodesic metric is a high computational cost of the algorithms based on it. The main goal of this ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- The American Mathematical Monthly
دوره 117 شماره
صفحات -
تاریخ انتشار 2010